Question

The locus of the points of intersection of the tangents to the circle $x=r \cos \theta, y=r \sin \theta$ at points whose parametric angles differ by $\pi / 3$, is

• A. $x^{2}+y^{2}=\frac{2}{3} r^{2}$
• B. $x^{2}+y^{2}=\frac{1}{3} r^{2}$
• C. $x^{2}+y^{2}=\frac{4}{3} r^{2}$
• D. None of these

Answer 1

Answer: (C)

All such points P satisfying the given condition will be equidistant from the origin $O$ (see fig.)

Hence the locus of $P$ will be a circle centred at the origin, having radius equal to
$OP =\frac{ r }{\cos \left(\frac{\pi}{6}\right)}=\frac{2 r }{\sqrt{3}}$
Therefore, equation of the
requieed locus is $ x^{2}+ y ^{2}=\frac{4}{3} r ^{2}$